To find out what the divisors of 24 are, as well as any whole number, a prime factorization is performed along with a few additional steps. It is a fairly short process and easy to learn.
When before decomposition into prime factors was mentioned, reference is being made to two definitions that are: factors and prime numbers.
The prime factorization of a number refers to rewriting that number as a product of prime numbers, where each of them is called a factor.
For example, 6 can be written as 2 × 3, therefore 2 and 3 are the prime factors in the decomposition.
The answer to this question is YES, and this is ensured by the following theorem:
Fundamental Theorem of Arithmetic: every positive integer greater than 1 is either a prime number or a single product of prime numbers except for the order of the factors.
According to the previous theorem, when a number is prime it has no decomposition.
Since 24 is not a prime number then it must be a product of prime numbers. To find them, the following steps are carried out:
-Divide 24 by 2, which gives a result of 12.
-Now divide 12 by 2, which gives 6.
-Divide 6 by 2 and the result is 3.
-Finally 3 is divided by 3 and the final result is 1.
Therefore, the prime factors of 24 are 2 and 3, but the 2 must be raised to the power 3 (since it was divided by 2 three times).
So 24 = 2³x3.
We already have the decomposition into prime factors of 24. It only remains to calculate its divisors. Which is done by answering the following question: What relationship do the prime factors of a number have with their divisors?
The answer is that the divisors of a number are their separate prime factors, along with the various products between them..
In our case, the prime factors are 2³ and 3. Therefore 2 and 3 are divisors of 24. From what has been said before, the product of 2 by 3 is a divisor of 24, that is, 2 × 3 = 6 is a divisor of 24.
There is more? Yes of course. As stated before, the prime factor 2 appears three times in the decomposition. Therefore, 2 × 2 is also a divisor of 24, that is, 2 × 2 = 4 divides 24.
The same reasoning can be applied for 2x2x2 = 8, 2x2x3 = 12, 2x2x2x3 = 24.
The list that was formed before is: 2, 3, 4, 6, 8, 12 and 24. Are they all?
No. You must remember to add to this list the number 1 and also all the negative numbers corresponding to the previous list.
Therefore, all the divisors of 24 are: ± 1, ± 2, ± 3, ± 4, ± 6, ± 8, ± 12 and ± 24.
As said at the beginning it is a fairly easy process to learn. For example, if you want to calculate the divisors of 36, you decompose into prime factors.
As seen in the image above, the prime factorization of 36 is 2x2x3x3.
So the divisors are: 2, 3, 2 × 2, 2 × 3, 3 × 3, 2x2x3, 2x3x3, and 2x2x3x3. And also the number 1 and the corresponding negative numbers must be added.
In conclusion, the divisors of 36 are ± 1, ± 2, ± 3, ± 4, ± 6, ± 9, ± 12, ± 18 and ± 36.
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